Ask students to solve the four problems in Questions 1–4 of the Multiplication at the Zoo pages in the Student Guide. Each can be solved using multiplication. Allow students to solve the problems in any way that makes sense to them, encouraging them to possibly use a different strategy on each problem. Encourage them to work on the problems themselves, discussing their solutions with partners as needed.
While students are working, identify some students to share their strategies with the class. Try to select a variety of strategies. Ask selected students to record their strategies on pieces of chart paper and to name the strategy they used. Some students will use strategies they have learned (e.g., break apart) and others will invent strategies (e.g., using money). Sample strategies are shown in Figures 1–6 in the Content Note on the following pages.
Lead a whole-class discussion about the different strategies used to solve Questions 1–4. Ask the selected students to share the recorded solutions. Then post the chart paper around the classroom for later use in this lesson and in Lessons 9 and 10.
- Can someone show how to use [student's name]'s method to solve one of the other problems?
- How is [student's name]'s method similar to [another student's name]?
- How are they different?
- Which methods would you choose if you did not have paper and pencil?
If students rely on one strategy, challenge them to find a second method for each problem. In particular, encourage them to use mental math.
Question 4 is an example of a problem that calls for an estimate. Because it is not possible to know how many pounds of food the lions eat each day (the problem says between 8 to 9 pounds), and because the number of days in a month varies, an estimate is the best we can do to answer the question. Let students discuss the problem with a partner and decide how to solve it before discussing it as a class. Refer students to the When Do We Estimate? display from Lesson 6.
Children often invent their own strategies for solving multiplication problems. Other strategies are developed through conversations with others. Even after students have developed proficiency with paper-and-pencil algorithms, these invented methods are helpful for mental arithmetic and estimation. Allowing students to explore their own ideas and hear the ideas of their classmates helps them develop flexibility with numbers and computation.
In Unit 3, students practiced solving multiplication problems by breaking products into tens and ones. This is a powerful strategy and is the basis of the standard multiplication algorithm. However, it is not the only strategy. This lesson gives students an opportunity to solve problems in ways that are natural to them and to discuss the pros and cons of various methods. Students should have a collection of methods that they find useful for solving problems, and they should understand the methods described by others.
- Is it appropriate to estimate for Question 4? Why? (Yes, because we don't know exactly how many days in the month or how much the lions eat. Also, it asks “about how much . . .”)
- How did you estimate how much the lions ate? (Student strategies and answers will vary. A possible solution is to use 30 days for a month and then multiply 8 × 30 = 240 pounds and 9 × 30 = 270 pounds. Numbers in this range are reasonable estimates.)
Sample Multiplication Strategies
Research has identified three types of strategies students frequently use to solve 1-digit by 2-digit multiplication problems: complete-number strategies, partitioning strategies, and compensation strategies. Below we give sample solutions to Questions 1–4 to illustrate these strategies. See the Mathematics in this Unit section for more discussion.
In complete-number strategies, students do not break the number apart. They may solve 32 × 5 (Question 2) by adding 32 five times as in Figure 1 or by writing 32 five times and grouping terms as in Figure 2.
In partitioning strategies, students break the numbers into the sum of parts and then multiply. This strategy depends on the distributive property. Figure 3 shows three natural ways to break apart 26 to solve 26 × 3 (Question 1). Besides breaking a number into tens and ones as they did in Unit 3, it is also helpful to break a number so that one of the parts is 25 or 50, since multiples of 25 and 50 are easy to compute with mentally.
In compensation strategies, students adjust the numbers to make calculations easier. In Figure 4, the problem 49 × 7 (Question 3) is changed to the easier problem 50 × 7. Then 1 × 7 is subtracted to compensate for this change. Another compensation strategy is shown in Figure 5. There the problem 32 × 5 is solved by cutting one factor in half and doubling the other. This changes the problem so the task is to multiply by 10, which is easier than multiplying by 5.
Figure 6 shows solutions to 22 × 8 (Question 5A). In the first solution, a rectangle is sketched to show how 22 can be broken into 20 and 2. The second solution combines methods. Multiply 22 by 4 to change the problem to 88 × 2. Then, since it is easier to multiply by 90 than by 88, calculate 90 × 2 and subtract 2 × 2.
Refer students to the strategies recorded and posted on the chart paper displayed around the room to help them solve 28 × 7. To get students to talk about some other strategies, use the student samples above Question 5 in the Student Guide. The student samples present a break-apart strategy using 25 (as in Figure 3) and a compensation strategy (as in Figures 4 and 5).
Read and discuss each student sample.
For example, you might ask:
- Is Jerome's method similar to one of the strategies we have already posted?
- Retell how Jerome solved the problem.
- Show us how to solve 49 × 7 using Jerome's method. (A simpler problem is 50 × 7. 50 × 7 = 350. Subtract one 7 for 343.)
- Do you think Jerome's strategy makes sense for any of the other problems?
If Jerome's strategy or Tanya's strategy are not similar to the ones you have already posted, ask a student to record these strategies on chart paper by solving any one of Questions 1×4. Name the strategy so you can refer to it easily.
After discussing the strategies, ask:
- Which methods would you choose if you didn't have paper and pencil?
Once all these strategies are posted, ask students to solve the problems in Question 5 using a method similar to Jeromes, Tanya's, or a strategy from the class.
